System Of Hyperbolic Partial Differential Equations Research Articles

Polynomial Fuzzy Observer-Based Feedback Control for Nonlinear Hyperbolic PDEs Systems.

This article explores the observer-based feedback control problem for a nonlinear hyperbolic partial differential equations (PDEs) system. Initially, the polynomial fuzzy hyperbolic PDEs (PFHPDEs) model is established through the utilization of the fuzzy identification approach, derived from the nonlinear hyperbolic PDEs model. Various types of state estimation and controller design problems for the polynomial fuzzy PDEs system are discussed concerning the state estimation problem. To investigate the relaxed stability problem, Euler's homogeneous theorem, Lyapunov-Krasovskii functional with polynomial matrices (LKFPM), and the sum-of-squares (SOSs) approach are adopted. The exponential stabilization condition is formulated in terms of the spatial-derivative-SOSs (SD-SOSs). Additionally, a segmental algorithm is developed to find the feasible solution for the SD-SOS condition. Finally, a hyperbolic PDEs system and several numerical examples are provided to illustrate the validity and effectiveness of the proposed results.

Prescribed‐time constrained tracking control of a class of 2 × 2 hyperbolic PDE systems with actuator dynamics

AbstractThis article studies the problem of prescribed‐time constrained tracking control for a class of hyperbolic partial differential equation (PDE) systems with actuator dynamics, which are described by a set of nonlinear ordinary differential equations (ODEs). Since the control input only appears in the ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite challenging. The most important is that for the control of such ODE controlled PDE systems we mainly make the following two contributions: (1) the controlled output of the PDE system tracks the reference signal within the prescribed time; (2) the controlled output and all the actuator states are constrained. It is the first time that such a prescribed‐time constrained tracking control problem is addressed for the PDE‐ODE coupled system considered in this article. Through rigorous theoretical proof, it is demonstrated that all the system states and control signals are bounded and sufficiently continuous by configuring appropriate design parameters. Finally, the performance is investigated via numerical simulation.

Concurrent learning adaptive boundary observer design for linear coupled hyperbolic partial differential equation systems

This paper proposes a novel concurrent learning-based adaptive boundary observer designed to tackle the joint estimation problem of system states and unknown parameters for a class of hyperbolic partial differential equation systems under the circumstance of unsatisfied persistent excitation conditions. By leveraging concurrent learning technique, an adapted data points selection algorithm is employed concurrently with current data to construct the adaptation law of unknown parameters, which overcomes the limitations imposed by persistent excitation conditions and ensures exponential convergence of estimation errors of unknown parameters under finite excitation conditions. Furthermore, combining the ideas of backstepping and swapping filters on the basis of uncertain estimation, a concurrent learning-based adaptive boundary observer is designed, accompanied by rigorous theoretical analysis and proofs to demonstrate its capability of achieving exponential convergence in estimation. Finally, the corresponding illustrative simulations are given to show the effectiveness of the proposed methodology.

Numerical analysis of a chaotic coupled hyperbolic PDE system for secure image encryption

This paper focuses on the interest of using chaotic hyperbolic Partial Differential Equations (PDE) systems synchronization for secure image communication. Such systems offer a promising alternative to finite dimensional chaotic systems with enlarged key space, efficient computation, and more complex dynamics. Here, we modify the classical chaotic hyperbolic PDE system by adding integral boundary couplings and potential terms. The chaotic behaviour of the resulting system is observed in numerical simulations. Adding such couplings allows to enhance the security of the cryptosystem. The synchronization of receiver and transmitter chaotic systems is achieved by the use of a chaotic observer designed using the backstepping methodology. The initial image can then be encrypted into a chaotic signal on the transmitter side before being sent to the receiver side. Using observational data, the receiver can then reconstruct the initial image. Numerical illustrations complete the paper.

Adaptive constrained tracking control for dynamical actuator driven linear 2 × 2 hyperbolic PDE systems with nonlinear uncertainties

The paper develops an adaptive constrained tracking control technique for a class of 2 × 2 hyperbolic partial differential equation (PDE) systems with boundary actuator dynamics, which are described by a set of ordinary differential equations (ODEs) in the presence of unknown parametric nonlinearities. Since the control input only appears in the uncertain ODE subsystem rather than directly on the boundary of PDE subsystem, the control task becomes quite difficult and the existing direct boundary control approaches are ineffective. Moreover, in this paper, a more challenging problem is considered such that the controlled output and the states of ODE actuators are constrained. To this end, by utilizing finite and infinite dimensional backstepping techniques, barrier Lyapunov functions (BLFs) and adaptive methods, a novel adaptive tracking control approach is proposed. It is the first time that such a constrained tracking control problem is addressed for the PDE-ODE coupled systems considered in this paper. On the basis of the presented method, the rigorous theoretical proof is provided to show that the PDE controlled output and all the states of the ODE actuator stay within the predefined compact sets. Finally, the results are illustrated via a comparative numerical simulation.

Adaptive Fuzzy Boundary Observer Design for Uncertain Linear Coupled Hyperbolic Partial Differential Equation Systems

Joint uncertainties and state estimation of a class of linear coupled hyperbolic partial differential equation systems in the presence of unstructured and structured uncertainties are studied in this paper. For unstructured uncertainties which are completely unknown, by employing Takagi-Sugeno fuzzy logic system to approximate the unstructured uncertainties, a novel adaptive fuzzy boundary observer is developed to estimate both unknown system states as well as unknown weights in the fuzzy logic system, and the estimation errors are ultimately bounded. Therein, in the design of the proposed observer, a set of swapping filters and infinite dimensional backstepping technique are combined. On the other hand, for structured uncertainties that can be described in a concrete parameterized form, the proposed method can easily achieve the exact estimation of weights and states to their true values. The rigorous proof is provided to show that the ultimately bounded estimation errors for the case of unstructured uncertainties and the exponential convergent estimation errors for the case of structured uncertainties can be realized. Finally, three illustrative simulations are carried out to show the feasibility and effectiveness of the developed methods in this paper.

Stochastic finite volume method for uncertainty quantification of transient flow in gas pipeline networks

We develop a weakly intrusive framework to simulate the propagation of uncertainty in solutions of generic hyperbolic partial differential equation systems on graph-connected domains with nodal coupling and boundary conditions. The method is based on the Stochastic Finite Volume (SFV) approach, and can be applied for uncertainty quantification (UQ) of the dynamical state of fluid flow over actuated transport networks. The numerical scheme has specific advantages for modeling intertemporal uncertainty in time-varying boundary parameters, which cannot be characterized by strict upper and lower (interval) bounds. We describe the scheme for a single pipe, and then formulate the controlled junction Riemann problem (JRP) that enables the extension to general network structures. We demonstrate the method's capabilities and performance characteristics using a standard benchmark test network.

Fault-compensation-based boundary control for hyperbolic PDEs: An adaptive iterative learning scheme

In this paper, a fault-compensation-based boundary control strategy is developed for a class of distributed parameter systems expressed by second-order hyperbolic Partial Differential Equations (PDEs), in the case of actuator faults. Considering multiplicative actuator faults and additive actuator faults simultaneously, we put forward a hierarchical fault compensation scheme, i.e., an adaptive iterative learning boundary control technique. Lyapunov-Like analysis method and a variation of Wirtinger’s inequality are used to design a boundary control scheme with an adaptive law, guaranteeing the asymptotical stability of the closed-loop hyperbolic PDE system with multiplicative faults. An iterative learning term is proposed to boost the performance of the system under both multiplicative faults and additive faults, and a composite energy function is used in analysis. With the proposed fault-compensation-based adaptive iterative learning boundary control technique, multiplicative faults are redeemed towards the time horizon and additive faults are compensated towards the iteration horizon. An active acoustic noise reduction process is presented to illustrate the merit and effectiveness of the designed adaptive iterative learning boundary control technique.

Output‐feedback boundary adaptive fault‐tolerant control for scalar hyperbolic partial differential equation systems with actuator faults

SummaryThis article studies the output‐feedback adaptive fault‐tolerant boundary control problem for scalar hyperbolic partial differential equation systems with actuator faults. All the coefficients of the controlled plant are unknown, and two types of actuator faults, that is, multiplicative faults and additive faults, are considered simultaneously. For the state estimation problem, two filters are constructed, based on which an observer is obtained to estimate the system state. A parametric model is established for actuator faults, based on which the parameter updating laws of gradient type are then developed to identify actuator faults and to estimate the unknown system coefficients. With the observer and the parameter updating laws, an output‐feedback adaptive fault‐tolerant boundary control law is developed via infinite‐dimensional backstepping method. The boundness of all the signals involved in the control design is guaranteed and the convergence of system states is also confirmed. Finally, the simulation results are given to testify the effectiveness of the proposed control scheme.

Finite‐time parameter estimation based boundary control for a class of 2×2 hyperbolic partial differential equation systems with uncertain transport speeds

AbstractThis article investigates the boundary control problem for a class of hyperbolic partial differential equation systems with uncertain transport speeds, in which both state feedback and output feedback cases are considered. The least‐square method is utilized to achieve the finite‐time parameter estimation based on the parametric models. During parameter estimation, proper actuation signals are designed via contradiction method to avoid the singular problems. With the estimated parameters, control laws are designed by the backstepping method for both state‐feedback case and output‐feedback case. Finally, simulation studies are provided to illustrate the efficiency of our results.

Fast hyperbolic relaxation elliptic solver for numerical relativity: Conformally flat, binary puncture initial data

We introduce nrpyelliptic, an elliptic solver for numerical relativity (NR) built within the nrpy+ framework. As its first application, nrpyelliptic sets up conformally flat, binary black hole (BBH) puncture initial data (ID) on a single numerical domain, similar to the widely used twopunctures code. Unlike twopunctures, nrpyelliptic employs a hyperbolic relaxation scheme, whereby arbitrary elliptic partial differential equations (PDEs) are trivially transformed into a hyperbolic system of PDEs. As consumers of NR ID generally already possess expertise in solving hyperbolic PDEs, they will generally find nrpyelliptic easier to tweak and extend than other NR elliptic solvers. When evolved forward in (pseudo)time, the hyperbolic system exponentially reaches a steady state that solves the elliptic PDEs. Notably nrpyelliptic accelerates the relaxation waves, which makes it many orders of magnitude faster than the usual constant wave speed approach. While it is still $\ensuremath{\sim}12\mathrm{x}$ slower than twopunctures at setting up full-3D BBH ID, nrpyelliptic requires only $\ensuremath{\approx}0.3%$ of the runtime for a full BBH simulation in the einstein toolkit. Future work will focus on improving performance and generating other types of ID, such as binary neutron stars.

Observer and output feedback control for nonlinear ordinary differential equation coupled to an under‐actuated transport partial differential equation

AbstractIn this article, the under‐actuation problem for a class of first order hyperbolic partial differential equation system coupled to a nonlinear ordinary differential equation is considered. An anticollocated observer design and an appropriate output feedback control is developed based on the backstepping methodology. The stability conditions for the nonlinear coupled system are proposed using the Lyapunov theory and are formulated in terms of the LMI constraints. Further, we have extended the stability results and backstepping control design for the fractional‐order coupled system. The stability conditions and the control law proposed in the work are evaluated considering a numerical simulation.

Multi-directional continuous traffic model for large-scale urban networks

In this paper, we propose a new multi-directional two-dimensional continuous model for urban traffic. It is called the NEWS model, since it represents a system of four partial differential equations (PDEs) that describe propagation of vehicle density in cardinal direction layers: North, South, West and East. The NEWS model can be applied to predict traffic evolution on a general urban network of arbitrary size by knowing only its boundary flows, as well as its topology and infrastructure parameters such as roads speed limits, number of lanes and capacities. The flux direction is retrieved from turning ratios at intersections, which is then aggregated in four direction layers using geometrical projection matrices. We show its formal derivation step-by-step from the classical Cell Transmission Model at one intersection. We then show that the NEWS model is a hyperbolic PDE system that corresponds to a conservation law with bounded densities. The model prediction performance is validated using synthetic data from the microsimulator Aimsun. Finally, the model is also validated using real data collected from a network of sensors installed in Grenoble (a city in France).

Backstepping Control of a Hyperbolic PDE System with Zero Characteristic Speed

In this paper, we study the single-input boundary feedback stabilization of 3 × 3 linear hyperbolic partial differential equations (PDEs) with two counterconvecting PDEs and the third one with zero characteristic speed. We design a full-state backstepping controller which exponentially stabilizes the origin in the L2 sense. The zero transport velocity makes the previous backstepping designs inapplicable (their application would result in a controller with infinite gains). To employ backstepping in the presence of zero speed, we use an invertible Volterra transformation only for the PDEs with nonzero speeds, leaving the state of the zero-speed PDE unaltered in the target system, but making the target zero-speed PDE input-to-state stable with respect to the decoupled and stable counterconvecting nonzero-speed PDEs. In addition to achieving stabilization, we produce an explicit bound on the rate of convergence of the target system by a method of successive approximations and the use of Laplace transform. Simulation results are presented to illustrate the effectiveness of the proposed control design.

Chaotic Dynamics of a 2D Hyperbolic PDE with the Boundary Conditions of Superlinear Type

The study of the chaotic dynamics of partial differential equation (PDE) system has become the focus of dynamical system. While important progress has been made for the research of chaos theory of one-dimensional wave PDE, the understanding of chaotic vibrations for two-dimensional (2D) hyperbolic PDE is still incomplete. This paper is concerned with a 2D hyperbolic PDE system governed by a linear equation with two suplinear boundary conditions. Based on the chaotic mapping theory and method of characteristics, sufficient conditions for the existence of nonisotropic chaotic vibrations are obtained for such system under three different system parameters. In addition, three numerical examples are provided to illustrate the effectiveness of our theoretical results.

Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equation fit within this unified framework. Secondly, we introduce the ongoing project SCRIMP (Simulation and Control of Interactions in Multi-Physics) developed for the numerical simulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS open-source computing platform for the finite element spatial discretization. Finally, we illustrate how to solve the considered model problems within this framework by carefully explaining the methodology. As additional support, companion interactive Jupyter notebooks are available.

Output feedback control of two-lane traffic congestion

This paper develops output feedback boundary control to mitigate traffic congestion of an unidirectional two-lane freeway segment. The macroscopic traffic dynamics are described by the Aw–Rascle–Zhang (ARZ) model respectively for both the fast and slow lanes. The traffic density and velocity of each of the two lanes are governed by coupled 2 × 2 nonlinear hyperbolic partial differential equations (PDEs). Lane-changing interactions between the two lanes lead to exchanging source terms between the two pairs second-order PDEs. Therefore, we are dealing with 4 × 4 nonlinear coupled hyperbolic PDEs. Based on driver’s preference for the slow and fast lanes, a reference system of lane-specific uniform steady states in congested traffic is chosen. To stabilize traffic densities and velocities of both lanes to the steady states, two distinct variable speed limits (VSLs) are applied at outlet boundary, controlling the traffic velocity of each lane. Using backstepping transformation, we map the coupled heterodirectional hyperbolic PDE system into a cascade target system, in which the traffic oscillations are damped out through actuation of the velocities at the downstream boundary. Two full-state feedback boundary control laws are developed. We also design a collocated boundary observer for state estimation with sensing of the densities at the outlet. Output feedback boundary controllers are obtained by combining the collocated observer and full-state feedback controllers. The finite time convergence to equilibrium is achieved and a performance improvement in fuel consumption, drivers’ comfort and total travel time is demonstrated with the proposed output feedback controllers for some parameter choice, compared with the open-loop system. The proposed output feedback design is also validated for different congested traffic scenarios and compared with one-lane backstepping and PI control designs.

Observer-based sliding mode control for stochastic hyperbolic PDE systems with quantized output signal

In this note, a sliding mode control scheme is firstly developed for a class of stochastic hyperbolic partial differential equation (PDE) systems. Initially, it is assumed that the considered systems’ states are unavailable for the controller, thus a new state observer is designed. Furthermore, to save the communication resources in network, the system’s measure output signal is quantized before transmission, based on which a novel sliding mode controller is constructed to enhance the system’s robustness. Then, some new criteria that guarantee the systems’ stability and H∞ disturbance attenuation performance are established in terms of linear matrix inequality. In the end, two simulation studies are presented to demonstrate the effectiveness of the developed control strategy.

Simulation of THz Oscillations in Semiconductor Devices Based on Balance Equations

Instabilities of electron plasma waves in high-mobility semiconductor devices have recently attracted a lot of attention as a possible candidate for closing the THz gap. Conventional moments-based transport models usually neglect time derivatives in the constitutive equations for vectorial quantities, resulting in parabolic systems of partial differential equations (PDE). To describe plasma waves however, such time derivatives need to be included, resulting in hyperbolic rather than parabolic systems of PDEs; thus the fundamental nature of these equation systems is changed completely. Additional nonlinear terms render the existing numerical stabilization methods for semiconductor simulation practically useless. On the other hand there are plenty of numerical methods for hyperbolic systems of PDEs in the form of conservation laws. Standard numerical schemes for conservation laws, however, are often either incapable of correctly handling the large source terms present in semiconductor devices due to built-in electric fields, or rely heavily on variable transformations which are specific to the equation system at hand (e.g. the shallow water equations), and can not be generalized easily to different equations. In this paper we develop a novel well-balanced numerical scheme for hyperbolic systems of PDEs with source terms and apply it to a simple yet non-linear electron transport model.

Duality-based optimal compensator for boundary control hyperbolic PDEs system: Application to a tubular cracking reactor

This paper is devoted to the design of an optimal stabilizing compensator for a boundary control distributed parameter system that is described by a set of hyperbolic partial differential equations (PDEs). The standard reformulation of a boundary control system is adopted here to write the system under a regular infinite-dimensional linear system. A finite-dimensional boundary optimal controller is designed based on the linear quadratic technique and the corresponding operator Riccati equation. On the other hand, a Luenberger observer is designed based on the duality between the control and the estimation problems. Combination of the designed controller and observer is performed to construct a stabilizing compensator. A case study of tubular cracking chemical reactor is used to test the performances of the developed algorithm.